If `y = log [cos (x^(5))]` then find ` (dy)/(dx)`

Answer

To find the derivative of \( y = \log(\cos(x^5)) \), we will apply the chain rule of differentiation. Here’s the step-by-step solution:

### Step 1: Identify the outer and inner functions
We have:
- Outer function: \( f(u) = \log(u) \) where \( u = \cos(x^5) \)
- Inner function: \( g(x) = \cos(x^5) \)

### Step 2: Differentiate the outer function
Using the chain rule, the derivative of \( y \) with respect to \( x \) is given by:
\[ \frac{dy}{dx} = \frac{d}{du}[\log(u)] \cdot \frac{du}{dx} \]
The derivative of \( \log(u) \) is:
\[ \frac{d}{du}[\log(u)] = \frac{1}{u} \]
Thus, we have:
\[ \frac{dy}{dx} = \frac{1}{\cos(x^5)} \cdot \frac{du}{dx} \]

### Step 3: Differentiate the inner function
Now we need to find \( \frac{du}{dx} \):
\[ u = \cos(x^5) \]
Using the chain rule again, we differentiate \( \cos(x^5) \):
\[ \frac{du}{dx} = -\sin(x^5) \cdot \frac{d}{dx}(x^5) \]
The derivative of \( x^5 \) is:
\[ \frac{d}{dx}(x^5) = 5x^4 \]
So, we have:
\[ \frac{du}{dx} = -\sin(x^5) \cdot 5x^4 \]

### Step 4: Combine the results
Substituting \( \frac{du}{dx} \) back into the equation for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{1}{\cos(x^5)} \cdot (-\sin(x^5) \cdot 5x^4) \]
This simplifies to:
\[ \frac{dy}{dx} = -\frac{5x^4 \sin(x^5)}{\cos(x^5)} \]

### Step 5: Final result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = -5x^4 \tan(x^5) \]